Coevolutionary dynamics of opinion propagation and social balance: The key role of small-worldness

The propagation of various opinions in social networks, which influences human interrelationships and even social structure, and hence is a most important part of social life. We have incorporated social balance into opinion propagation in social networks are influenced by social balance. The edges in networks can represent both friendly or hostile relations, and change with the opinions of individual nodes. We introduce a model to characterize the coevolutionary dynamics of these two dynamical processes on Watts-Strogatz (WS) small-world network. We employ two distinct evolution rules (i) opinion renewal; and (ii) relation adjustment. By changing the rewiring probability, and thus the small-worldness of the WS network, we found that the time for the system to reach balanced states depends critically on both the average path length and clustering coefficient of the network, which is different than other networked process like epidemic spreading. In particular, the system equilibrates most quickly when the underlying network demonstrates strong small-worldness, i.e., small average path lengths and large clustering coefficient. We also find that opinion clusters emerge in the process of the network approaching the global equilibrium, and a measure of global contrariety is proposed to quantify the balanced state of a social network.